Electron diffraction

Ewald Sphere construction for Transmission Electron Diffraction, showing two of the Laue zones and the excitation error

What one sees in an electron diffraction pattern depends upon the sample and also the energy of the electrons. The electrons need to be considered as waves, which can represented by writing the wavefunction as:

${\displaystyle \psi (r)=exp(2\pi ik.r)}$

for a position ${\displaystyle r}$. The vector ${\displaystyle k}$ is called the wavevector, and the form above as a "plane wave" as the term inside the exponential is constant on the surface of a plane. The vector ${\displaystyle k}$ is also what is used when one draws ray diagrams.

For most cases the electrons are travelling at a respectable fraction of the speed of light, so rigorously they need to be considered[20] in terms of the Dirac equation, which as spin does not normally matter can be reduced to the Klein–Gordon equation. Fortunately one can side-step this and use a non-relativistic approach based around the Schrödinger equation. Following Fujiwara[21] and Howie,[22] the relationship between the total energy of the electrons and the wavevector is written as:

${\displaystyle E=h^{2}k^{2}/2m^{eff}}$ with ${\displaystyle m^{eff}=m_{0}+eE/2c^{2}}$

where ${\displaystyle h}$ is Planck's constant, ${\displaystyle m^{eff}}$ is an effective mass adjusted to cancel out the relativistic terms for the energy ${\displaystyle E}$ with ${\displaystyle c}$ the speed of light and ${\displaystyle m_{0}}$ the rest mass of the electron. The wavelength of the electrons is ${\displaystyle 1/k}$, and can range from about 0.1 nanometers, about the size of an atom, down to a thousandth of that.

The magnitude of the interaction of the electrons with a material scales as what is called the relativistic interaction constant[23] ${\displaystyle 2\pi m^{eff}e/h^{2}k}$, where ${\displaystyle e}$ is the electron charge. While the wavevector increases as the energy increases, the change in the effective mass somewhat compensates this so even at the very high energies often used in electron diffraction there is still significant interactions.

The electrons interact with the Coulomb potential, which for a crystal can be considered in terms of a Fourier series (see for instance Ashcroft and Mermin) that is

${\displaystyle V(r)=\sum V_{g}exp(2\pi ig.r)}$

with ${\displaystyle g}$ a Reciprocal lattice vector and ${\displaystyle V_{g}}$ the Fourier coefficient of the potential. The reciprocal lattice vector is often referred to in terms of Miller indices ${\displaystyle (h,k,l)}$, by which one means that it is a sum of the individual reciprocal lattice vectors ${\displaystyle A,B,C}$ with integer ${\displaystyle h,k,l}$ in the form:

${\displaystyle g=hA+kB+lC}$

The contribution from the ${\displaystyle V_{g}}$ needs to be combined with what is called the shape function (e.g.[24][25][23]), which is the Fourier transform of the shape of the object. If, for instance, the object is small in one dimension then the shape function extends far in that direction in the Fourier transform—a reciprocal relationship.

Around each reciprocal lattice point one then has this shape function.[26][27] How much intensity one has then depends upon the intersection of the Ewald sphere, that is energy conservation, and the shape function around each reciprocal lattice point. For transmission electron diffraction the samples used tend to be relatively thin, so most of the shape function is along the direction of the electron beam. For both Low-energy electron diffraction[28] and RHEED[29] the shape function is mainly normal to the surface of the sample. In Low-energy electron diffraction this mainly leads (a simplification) to back-reflection of the electrons leading to spots, whereas in Reflection high-energy electron diffraction the electrons reflect off the surface at a small angle and typically yield diffraction patterns with streaks. By comparison, with both x-ray and neutron diffraction the scattering is significantly weaker, so typically one needs much larger crystals in which case the shape function shrinks to just around the reciprocal lattice points, leading to simpler Bragg's law diffraction.

For all cases, when the reciprocal lattice points are close to the Ewald sphere the intensity tends to be higher; when they are far away they tend to be smaller. The set of diffractions spots near to the direction of the incident beam are called the zero-order Laue zone (ZOLZ). One can also have intensities further out from reciprocal lattice points which are in a higher layer. The first of these is called the First order Laue zone (FOLZ); the series is called by the generic name Higher order Laue zone (HOLZ).

In Kinematical theory an approximation is made that the electrons are only scattered once,[23] and one normally make life simpler by also assuming a simple, flat sample. Going a little further, for transmission electron diffraction one typically assumes a constant thickness ${\displaystyle t}$, what is called the Column Approximation (e.g. references[26][30] and further reading). The intensity for each diffraction spot is then proportional to

${\displaystyle I_{g}=|F_{g}sin(\pi ts_{z})/(\pi s_{z})|^{2}}$

where ${\displaystyle F_{g}}$ is the structure factor:

${\displaystyle F_{g}=\sum _{j=1}^{N}f_{j}\mathrm {e} ^{(2\pi ig.r_{j}-Tg^{2})}}$

the sum being over all the atoms in the unit cell with ${\displaystyle f_{j}}$ the form factors,[2] ${\displaystyle g}$ the reciprocal lattice vector and ${\displaystyle T}$ is a simplified form of the Debye–Waller factor.

Of some importance, the intensity in transmission electron diffraction oscillates as a function of thickness, which can be confusing; one can similarly have complications due to these and defects within kinematical theory.[31] If a diffraction spots is strong it could be because it has a larger structure factor, or it could be because the combination of thickness and excitation error is "right". Similarly the observed intensity can be small, even though the structure factor is large. This can complicate interpretation of the intensities. By comparison, these effects are much smaller in X-ray diffraction or Neutron diffraction because they interact with matter far less and often one can simply use Bragg's law.

This form is a reasonable first approximation within about 20% in many cases, but one has to use much more accurate forms including multiple scattering of the electrons to properly understand the intensities. These approaches are called dynamical diffraction.

While kinematical diffraction is adequate to understand the geometry of where one has diffraction spots, it does not correctly give the intensities and has a number of other limitations. For a more complete approach one has to include multiple scattering of the electrons using methods that date back to the early work of Bethe in 1928.[18] Even at very high energies this is needed. The main components of this are:

• Include the scattering back into the incident beam and all others, not just single scattering.[18]
• Include at least semi-empirically the role of inelastic scattering by an imaginary component of the potential,[32][33][34] also called an "optical potential".[35]
• Use higher-order approaches to calculate the intensities such as Multislice[36][37] or matrix methods[38][39] which are called Bloch-wave approaches or muffin tin approaches.[40] With these diffraction spots which are not present in kinematical theory can be present, e.g.[41]
• Include effects due to local variations due to elastic strain and defects, and also what Lindhard called the string potential.[42]
• Include, for Transmission electron microscopy, effects due to variations in the thickness of the sample and the normal to the surface.[23]
• Include, for both Low-energy electron diffraction[43] and Reflection high-energy electron diffraction[44][45] effects due to the presence of surface steps, Surface reconstructions and other atoms at the surface.
• Include, for Low-energy electron diffraction, more careful analyses of the potential because contributions from exchange terms can be important.[46]

Crystallographic planes in [0 1 1] Mg cell with corresponding diffraction spots and Kikuchi lines simulated in CrysTBox

Kikuchi lines,[47][27] first observed in 1928,[48] are linear features created by electrons scattered both inelastically and elastically. As the electron beam interacts with matter, the electrons are diffracted via elastic scattering, and also scattered inelastically losing part of their kinetic energy. These occur simultaneously, and one cannot separate them – the Copenhagen interpretation. These electrons form Kikuchi lines which provide information on the orientation.[49]

Kikuchi lines come in pairs forming Kikuchi bands, and are indexed in terms of the crystallographic planes they are connected to. The angular width of the band is equal to the diffraction vector ${\displaystyle g}$, so in the diffraction pattern, the width of ${\displaystyle (hkl)}$ band will be equal to the distance between transmitted beam and ${\displaystyle (hkl)}$ diffraction spot. The position of Kikuchi bands is fixed with respect to each other and the orientation of the sample, but not against the diffraction spots or the direction of the incident electron beam. As the crystal is tilted, the bands move on the diffraction pattern.[49] Since the position of Kikuchi bands is quite sensitive to crystal orientation, they can be used to fine-tune a zone-axis orientation or determine crystal orientation with significantly higher accuracy than what is feasible with a spot diffraction analysis. They can also be used for navigation when changing the orientation between zone axes connected by some band. For those purposes, Kikuchi maps are available.[1]

Diffraction patterns with different crystallinity and beam convergence. From left: spot diffraction, CBED, ring diffraction

Electron diffraction in a Transmission electron microscopy (TEM) is a versatile technique which exploits the ability to form a very controlled electron beam using complex electron optics. Different types of diffraction experiments provide information such as measurement of the crystal lattice constants, information on crystal defects, charge variations, symmetries, and it can be used solve an unknown crystal structure. As a general overview see the classic text by Edgerton,[50] as well as the recommended reading.

It is common to combine it with other tools available in the microscope. Among other methods, TEM can provide magnified images using just selected diffraction beams, high-resolution images[51][52] showing the atomic structure, chemical analysis through energy-dispersive X-ray spectroscopy,[53] investigations of electronic structure and bonding through electron energy loss spectroscopy,[54] and studies of the mean inner potential through electron holography;[55] this list is not exhaustive. Compared to another widely used material characterization technique, X-ray diffraction, TEM analysis is significantly more localized and can be used to obtain information from tens of thousands of atoms to just a few or even single atoms.

Imaging scheme of magnetic lens (center) with magnified image (left) and diffraction pattern (right) formed in back focal plane

In TEM, the electron beam passes through a thin film of the examined material. Before and after its interaction with the sample, the beam is manipulated by various elements of the Electron optics[56] including magnetic lenses, deflectors and apertures;[57] these act on the electrons very much like how glass lenses focus and control light. Optical elements above the sample are used to control the incident beam which can range from a wide and parallel beam to one which is a converging cone and can be smaller than an atom, 0.1 nm. As it interacts with the sample, part of the beam is diffracted and part is transmitted through the sample without changing its direction. Which part? This one can never say as electrons are everywhere until they are detected according to the Copenhagen interpretation.

Below the sample, the beam is controlled by another set of magnetic lens and apertures. Each set of initially parallel rays is focused by the first lens Objective (optics) to a certain point in the back focal plane of the first lens, forming a spot. The location of these spots is related to the interplanar distance in the sample. Other lenses below the sample can be used to produce a magnified image of the spots for all the different directions that the electrons leave the sample, a diffraction pattern. Alternatively one can also form a magnified image of the sample. Modern microscopes allows one to switch between the imaging and diffraction mode by pressing a single button, which makes diffraction data easily available and accessible.[1] In the sections below the focus is on what one has when collecting a magnified diffraction pattern; for other information see the pages on Transmission electron microscopy and Scanning transmission electron microscopy.

Single-crystalline SADP automatically interpreted with CrysTBox software.

The simplest diffraction technique in TEM is selected area (electron) diffraction (SAED) where the incident beam is wide and close to parallel. To select a particular region of interest from which the diffraction is collected, a selected area aperture is used. These apertures are typically thin foils of a heavy metal such as Tungsten[57] which has a number of small holes in it. It is located below the sample, and can be positioned so that it only allows some portion of the electron beam to pass, blocking the rest. This way diffraction information can be limited to, for instance, individual crystallites. Unfortunately the method is limited by the spherical aberration of the objective lens, so is only accurate for large grains with tens of thousands of atoms or more; for smaller regions a focused probe is needed..

If a parallel beam is used to acquire a diffraction pattern from a single-crystal, the resulting image is similar to a two-dimensional projection of the crystal reciprocal lattice. From this one can determine interplanar distances and angles and in some cases crystal symmetry, particularly when the electron beam is down a major zone axis, see for instance the database by Jean-Paul Morniroli.[58] In combination with modern automated analytical software such as CrysTBox, SAED can be used for a quantitative analysis with reasonable precision.[59] However, projector lens aberrations such as Barrel Distortion as well as dynamical diffraction effects (e.g.[60]) cannot be ignored. For instance, certain diffraction spots which are not present in x-ray diffraction can appear,[58] for instance Gjonnes-Moodie extinction conditions.[41] Hence x-ray diffraction remains the preferred method for precise lattice parameter measurements.

Diffraction pattern of magnesium simulated using CrysTBox for various crystal orientations.

If the sample is tilted relative to the electron beam, different sets of crystallographic planes contribute to the pattern yielding different types of diffraction patterns, approximately different projections of the reciprocal lattice[58]. This can be used to determine the crystal orientation, which in turn can be used to set the orientation needed for a particular experiment, for instance to determine the misorientation between adjacent grains or crystal twins.[61][59] Furthermore a series of diffraction patterns varying in tilt can be acquired and processed using a diffraction tomography approach. There are then ways to combine this with direct methods algorithms using electrons[62][63] and other methods such as charge flipping,[64] or automated diffraction tomography.[65][66]

Apart from limiting the beam using the selected area aperture, localization can be achieved by condensing the incident beam into a narrow electron probe. This technique is called microprobe or nanoprobe diffraction or simply nanodiffraction. Probes of diameter smaller than 1 nm can be obtained.[1] At these extremes the probe dimension comes at the price of the beam being nearly parallel, even though microprobe can be achieved with parallel beam.[61] Compared to SAED, the resulting diffraction spots can be significantly broader, making a manual analysis less accurate. Accurate processing is possible with software,[67][59][68] significantly improving accuracy and repeatability of the results.

Relation between spot and ring diffraction illustrated on 1 to 1000 grains of MgO using simulation engine of CrysTBox. Corresponding experimental image can be seen below.

The character of the resulting diffraction pattern depends on whether the beam is diffracted by one single crystal or by a number of differently oriented crystallites, for instance in a polycrystalline material. If there are more contributing crystallites, the diffraction image becomes a superposition of individual crystal patterns. Ultimately, this superposition contains diffraction spots of all possible crystallographic plane systems in all possible orientations. This results in a pattern of concentric rings of discrete radii for the following reasons:

1. The discreteness of ring radii is because there are discrete spacings between various crystallographic planes in a given crystal system and therefore can only form diffraction spots at discrete distances from the transmitted beam.
2. For a large number of grains there are all possible orientations and therefore the diffraction spots are formed all around the transmitted beam (rings' center) at a distance (radius) related to a particular crystallographic plane.

Ring diffraction image of MgO as recorded (left) and processed with CrysTBox ringGUI (right). Corresponding simulated pattern can be seen above.

Textured materials can be recognized by a non-uniform distribution of intensity along the ring. Ring diffraction patterns can be also used to discriminate between nanocrystalline and amorphous phases[61] by careful analysis of the width of the diffraction rings. However, diffraction is not as sensitive to differentiating between very small grain polycrystalline materials and truly random order amorphous. Here High-resolution transmission electron microscopy and Fluctuation electron microscopy[69][70] can be more powerful, although this is still a topic of continuing development.

In simple cases there is only one grain or one type of material in the area used for collecting a diffraction pattern. However, often there is more than one. If they are in different areas then the diffraction pattern will be a combination.[3] In addition one can have one superimposed on top of another, in which case the electrons that go through the first are diffracted by the second. Electrons have so memory (like many of us), so after they have gone through the first grain and been diffracted, they traverse the second as if their current direction was that of the incident beam. This leads to diffraction spots which are the sum of those of the two (or even more) reciprocal lattices of the crystals, and can lead to quite complicated results. It can be difficult to know if one has a real and perhaps novel material, or just one where multiple crystals and diffraction is leading to odd results.

Schematic of the convergent beam electron diffraction technique. Adapted from W. Kossel & G. Möllenstedt, 1939.

In Convergent Beam Electron Diffraction (CBED), the incident electrons are focused in a converging cone-shaped beam with a crossover often located at the sample, although other methods exist. Unlike the parallel beam, the convergent beam is able to carry information from the sample volume, not just a two-dimensional projection available in SAED. With convergent beam there is also no need for the selected area aperture, as it is inherently site-selective since the beam crossover is positioned at the object plane where the sample is located.[49]

A CBED pattern consists of disks arranged exactly the same as the spots in SAED. Intensity within the disks, however, is not uniform, but represents dynamical diffraction effects and symmetries of the sample structure. Even though the zone axis and lattice parameter analysis based on disk positions does not significantly differ from SAED, the analysis of disks content is significantly more complex. Due to a number of contributing factors, simulation based on dynamical diffraction theory is often required. With appropriate analysis, however, CBED patterns can be used for indexation of the crystal point group, space group identification, measurement of lattice parameters, thickness or strain.[49]

The disk diameter can be controlled using the microscope optics and apertures. The larger is the angle, the broader the disks are with more features. If the angle is increased to significantly, the disks begin to overlap. This can be solved for instance via a large angle convergent electron beam diffraction (LACBED) where the sample is moved upwards or downwards. There are applications, however, where the overlapping disks are beneficial. Ronchigram is as an example. It is a CBED pattern, often but not always of an amorphous material, with many intentionally overlapping disks blended into one image providing information about the defects (optical aberration) of the electron optical system.[1]

Geometry of electron beam in precession electron diffraction. Original diffraction patterns collected by C.S. Own at Northwestern University[71]

Precession electron diffraction (PED), first developed by Vincent and Midgley in 1994,[72] is a specialized method to collect electron diffraction patterns in a transmission electron microscope (TEM), see the main page for more information. By rotating (precessing) a tilted incident electron beam around the central axis of the microscope, a PED pattern is formed which is effectively an integration over a collection of diffraction conditions. This produces a quasi-kinematical diffraction pattern that is more suitable[73] as input into direct methods algorithms using electrons[62][63] to determine the crystal structure of the sample. Because it avoids many dynamical effects it can also be used to better identify phases.[74]

4D scanning transmission electron microscopy (4D STEM) is a subset of scanning transmission electron microscopy (STEM) methods which utilizes a pixelated electron detector to capture a convergent beam electron diffraction (CBED) pattern at each scan location; see the main page for further information. This technique captures a 2 dimensional reciprocal space image associated with each scan point as the beam rasters across a 2 dimensional region in real space, hence the name 4D STEM. Its development was enabled by evolution in STEM detectors and improvements in computational power. The technique has applications in visual diffraction imaging, phase orientation and strain mapping, phase contrast analysis, among others; it has become very popular and rapidly evolving from about 2020 onwards.

The name 4D STEM is common in literature, however it is known by other names: 4D STEM EELS, ND STEM (N- since the number of dimensions could be higher than 4), position resolved diffraction (PRD), spatial resolved diffractometry, momentum-resolved STEM, "nanobeam precision electron diffraction", scanning electron nano diffraction, nanobeam electron diffraction, or pixelated STEM.[75] Most of these are the same, although there are instances such as momentum-resolved STEM where the emphasis can be very different.

Figure 1: LEED pattern of a Si(100) reconstructed surface. The underlying lattice is a square lattice, while the surface reconstruction has a 2x1 periodicity. Also seen is the electron gun that generates the primary electron beam; it covers up parts of the screen.

Low-energy electron diffraction (LEED) is a technique for the determination of the surface structure of single-crystalline materials by bombardment with a collimated beam of low-energy electrons (30–200 eV)[76] and observation of diffracted electrons as spots on a fluorescent screen; see the main page for more information and reference.[77][28] It has been used to solve a very large number of relatively simple surface structures of metals and semiconductors, plus cases with simple chemisorbants. For more complex cases transmission electron diffraction[78][79] or surface x-ray diffraction[80] have been used, often combined with scanning tunnelling microscopy and density functional theory calculations.

LEED may be used in one of two ways:

1. Qualitatively, where the diffraction pattern is recorded and analysis of the spot positions gives information on the symmetry of the surface structure. In the presence of an adsorbate the qualitative analysis may reveal information about the size and rotational alignment of the adsorbate unit cell with respect to the substrate unit cell.
2. Quantitatively, where the intensities of diffracted beams are recorded as a function of incident electron beam energy to generate the so-called I–V curves. By comparison with theoretical curves, these may provide accurate information on atomic positions on the surface at hand.

RHEED, see the main page for more information and reference,[29] is a technique used to characterize the surface of crystalline materials. RHEED systems gather information only from the surface layers of the sample, which distinguishes RHEED from other materials characterization methods that also rely on diffraction of high-energy electrons. Transmission electron microscopy, another common electron diffraction method, sample mainly the bulk of the sample due to the geometry of the system, although in special cases it can provide surface information.[81] Low-energy electron diffraction (LEED) is also surface sensitive, but LEED achieves surface sensitivity through the use of low energy electrons. The main uses of RHEED to date have been during thin film growth,[82] as the geometry is amenable to simultaneous collection of the diffraction data and deposition. It can, for instance, be used to monitor surface roughness during growth.

GED pattern of benzene.

Gas electron diffraction (GED) can be used to determine the geometry of molecules in gases. A gas carrying the molecules is exposed to the electron beam, which is diffracted by the molecules. Since the diffracting molecules are randomly oriented, the resulting diffraction pattern consists of concentric rings. The diffraction intensity is a sum of several components such as background, atomic intensity or molecular intensity.

In GED the diffraction intensities at a particular diffraction angle ${\displaystyle \theta }$ is described via so-called scattering variable defined as

${\textstyle |s|={\frac {4\pi }{\lambda }}\sin \left({\frac {\theta }{2}}\right).}$

The total intensity is then given as a sum of partial contributions:

${\displaystyle I_{\text{tot}}(s)=I_{a}(s)+I_{m}(s)+I_{t}(s)+I_{b}(s),}$

where ${\displaystyle I_{a}(s)}$ results from scattering by individual atoms, ${\displaystyle I_{m}(s)}$ by pairs of atoms and ${\displaystyle I_{t}(s)}$ by atom triplets. Intensity ${\displaystyle I_{b}(s)}$ corresponds to the background which, unlike the previous contributions, must be determined experimentally. The intensity of atomic scattering ${\displaystyle I_{a}(s)}$ is defined as

$${\displaystyle I_{a}(s)={\frac {K^{2}}{R^{2}}}I_{0}\sum _{i=1}^{N}|f_{i}(s)|^{2},}$$

where ${\displaystyle K=(8\pi ^{2}me^{2})/h^{2}}$, ${\displaystyle R}$ is the distance between the scattering object detector, ${\displaystyle I_{0}}$ is the intensity of the primary electron beam and ${\displaystyle f_{i}(s)}$ is the scattering amplitude of the i-th atom in the molecular structure in the experiment. ${\displaystyle I_{a}(s)}$ is the main contribution and easily obtained for known gas composition.

The most valuable information is carried by the intensity of molecular scattering ${\displaystyle I_{a}(s)}$, as it contains information about the distance between all pairs of atoms in the molecule, whether bonded or not. It is given by formula

${\displaystyle I_{m}(s)={\frac {K^{2}}{R^{2}}}I_{0}\sum _{i=1}^{N}\sum _{\stackrel {j=1}{i\neq j}}^{N}\left|f_{i}(s)\right|\left|f_{j}(s)\right|{\frac {\sin[s(r_{ij}-\kappa s^{2})]}{sr_{ij}}}e^{-(1/2l_{ij}s^{2})}\cos[\eta _{i}(s)-\eta _{i}(s)],}$

where ${\displaystyle r_{ij}}$ is the distance between two atoms, ${\displaystyle l_{ij}}$ is the mean square amplitude of vibration between the two atoms, ${\displaystyle \kappa }$ is the anharmonicity constant and ${\displaystyle \eta }$is a phase factor which is important for atomic pairs with very different nuclear charges. The summation is performed over all atom pairs. Atomic triplet intensity ${\displaystyle I_{t}(s)}$ is negligible in most cases. If the molecular intensity is extracted from an experimental pattern by subtracting other contributions, it can be used to match and refine a structural model against the experimental data.

EBSD pattern of silicon.

In a Scanning electron microscope the region near the surface is mapped using a electron beam that is scanned in a grid across the sample. A diffraction pattern is formed using electron backscatter diffraction (EBSD) with a camera inside the microscope.[83] A depth from a few nanometers to a few microns, depending upon the electron energy used, is penetrated by the electrons, some of which are diffracted backwards and out of the sample. As result of combined inelastic and elastic scattering, typical features in an EBSD image are Kikuchi lines. Since the position of Kikuchi bands is highly sensitive to the crystal orientation, EBSD data acquired can be used to determine the crystal orientation at particular locations on the sample. The data are processed by an automated software allowing to routinely generate two-dimensional orientation maps across the sample.[84][85] As the Kikuchi lines carry information about the interplanar angles and distances and, therefore, about the crystal structure, they can be also used for a phase identification or strain analysis.[86]